As noted above, this paper studies how to construct a confidence sequence for the mean of bounded random processes (with known support).
The most closely related work is [1]}, [2]}, [3]}, [4]}.
blackBuilding upon the idea of gambling to construct confidence sequences pioneered by [1]} and [2]},
[3]} examined several heuristic betting strategies and their implications in terms of confidence sequences.
Recently, [4]} proposed to apply the idea of [9]}'s universal portfolio (UP) blackmethod together with a certain regret-based lower bound.
They demonstrated an excellent empirical performance of the proposed algorithm “PRECiSE”, compared to the blackthen state-of-the-art methods [3]} especially in a small-sample regime, and illustrated that their regret-based-lower-bound approach admits an explicit analysis of the method.
We remark, however, that the proposed regret-based approach is rather intricate and does not naturally result in confidence intervals.
Moreover, the PRECiSE algorithm suffers the same order of time complexity as Cover's UP, which is quadratic in the sequence length.blackIn the very recent arXiv revision of [4]} on July 18, 2022 (https://arxiv.org/abs/2110.14099v2), the algorithm we refer to PRECiSE in this paper is renamed as PRECiSE-CO97, and a new algorithm dubbed as PRECiSE-CO96 was further proposed as a fast approximate version of Cover's UP with constant
per-round complexity.
See Section  for a comparison.
